not so sure HR couldn't correct this especially if your coverage was obtained through your company's cafeteria plan. if it was through a cafeteria - section 125 plan - the employer and hence employees risk having the plan disqualified unless no employee could obtain pre-tax health insurance. Now if you declined coverage and then realized you wanted it, then once open enrollment closed it became your problem.
the deduction as an after-tax expense would be as a medical expense. that mess-up will likely cost you more income taxes, due to the limitation on medical expenses and then the need to be able to itemize your deductions to get any benefit. there is also the social security and medicare taxes you paid.